What is the remainder when x² + 3x + 5 is divided by (x − 1)?

9. Substitute x = 1: 1 + 3 + 5.

p(a) = 0. What does this tell you?

(x − a) is a factor of p(x). That is the factor theorem.

Which value should be substituted to test whether (x + 3) is a factor?

−3. The value that makes x + 3 equal to 0.

What is the remainder when 2x³ − x + 1 is divided by (x + 1)?

0. p(−1) = −2 + 1 + 1 = 0, so (x + 1) is a factor.

Which value should be substituted to find the remainder on division by (2x − 1)?

1/2. 2x − 1 = 0 when x = 1/2.

The remainder and factor theorems · Cambridge IGCSE Additional Mathematics

In plain words

Dividing a polynomial by something like (x − 2) is slow. The remainder theorem gives the remainder in one line, by substituting a number. And when that remainder is zero, you have found a factor.

4 things to know

The remainder theorem: when a polynomial p(x) is divided by (x − a), the remainder is p(a).
For a divisor (ax − b), the remainder is p(b/a): use the value of x that makes the divisor zero.
The factor theorem: (x − a) is a factor of p(x) exactly when p(a) = 0.
These are used to find unknown coefficients: substitute, and set the result equal to the given remainder, or to 0 for a factor.

Worked example

p(x) = x³ + 2x² − 5x − 6. Find the remainder when p(x) is divided by (x − 1), and show that (x − 2) is a factor.

Remainder on division by (x − 1): p(1) = 1 + 2 − 5 − 6 = −8.
p(2) = 8 + 8 − 10 − 6 = 0.
Since p(2) = 0, (x − 2) is a factor.

Worked example

When 2x³ + ax² − 3x + 4 is divided by (x − 2), the remainder is 10. Find a.

p(2) = 16 + 4a − 6 + 4 = 14 + 4a.
14 + 4a = 10, so 4a = −4 and a = −1.

Tips and tricks

The sign flips: for (x − 2) substitute 2, and for (x + 3) substitute −3.
To "show that" something is a factor, write the substitution out in full and finish with "= 0, so it is a factor".

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

The remainder and factor theorems: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the remainder when x² + 3x + 5 is divided by (x − 1)?
- 5
- 7
- 9 (the answer)
- 3
Substitute x = 1: 1 + 3 + 5.
p(a) = 0. What does this tell you?
- (x + a) is a factor of p(x)
- (x − a) is a factor of p(x) (the answer)
- p(x) has no factors
- the remainder is a
That is the factor theorem.
Which value should be substituted to test whether (x + 3) is a factor?
- 3
- −3 (the answer)
- 0
- 1/3
The value that makes x + 3 equal to 0.
What is the remainder when 2x³ − x + 1 is divided by (x + 1)?
- 0 (the answer)
- 2
- −2
- 4
p(−1) = −2 + 1 + 1 = 0, so (x + 1) is a factor.
Which value should be substituted to find the remainder on division by (2x − 1)?
- 1
- 2
- 1/2 (the answer)
- −1/2
2x − 1 = 0 when x = 1/2.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 7 marks. Write your answers on paper, then check them.